### Book

## Stability, Control, Differential Games (SCDG2019) Proceedings of the International Conference devoted to the 95th anniversary of Academician N.N. Krasovskii, Yekaterinburg, Russia, 16–20 September 2019

In this lecture, the applications of the Pyragas time-delay feedback control technique and Leonov analytical approach for estimation of Lyapunov dimension and topological entropy in the framework of studying the Eden’s conjecture are discussed. The problem of reliable numerical computation of the mentioned dimension-like characteristics along the trajectories over large time intervals is demonstrated.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

The introduction describes the concept in the "hard"and "soft" sciences.

Der vorliegende Bank ist einer fuenf jaehrigen Projektarbeit zu Synergie-Konzepten. Synergie ist ein Schlüsselbegriff in Wissenschaft und Gesellschaft. Wie wird er historisch und gegenwärtig verwendet? Was zeichnet ihn als produktives Paradigma in interdisziplinären Forschungs- und Praxisfeldern aus? Als Modell einer holistischen Beschreibung der Wirklichkeit macht die synergetische Perspektive die aristotelische Einsicht fruchtbar, dass das Ganze mehr ist als bloß die Summe seiner Teile. Allgemeine Theorien des Zusammenwirkens (synérgeia) nehmen hier ihren Ausgangspunkt. Mit Blick auf kooperative Interaktionen und dynamische Strukturbildungen in Natur, Kunst und Gesellschaft untersuchen die Beiträge philosophie-, wissenschafts- und kulturgeschichtliche Konstellationen, in denen Synergie-Konzepte besondere Konjunktur haben, und fragen nach dem Zukunftspotenzial dieser transdisziplinären Denkfigur.

Sunspot number *WN* displays quasi-periodical variations that undergo regime changes. These irregularities could indicate a chaotic system and be measured by Lyapunov exponents. We define a functional l (an “irregularity index”) that is close to the (maximal) Lyapunov exponent for dynamical systems and well defined for series with a random component: this allows one to work with sunspot numbers. We compute l for the daily *WN* from 1850 to 2012 within 4-year sliding windows: l exhibit sharp maxima at solar minima and secondary maxima at solar maxima. This pattern is reflected in the ratio *R* of the amplitudes of the main vs secondary peaks. Two regimes have alternated in the past 150 years, R1 from 1850 to 1915 (large l and *R* values) and R2 from 1935 to 2005 (shrinking difference between main and secondary maxima, *R* values between 1 and 2). We build an autoregressive model consisting of Poisson noise plus an 11-yr cycle, and compute its irregularity index. The transition from R1 to R2 can be reproduced by strengthening the autocorrelation *a *of the model series. The features of the two regimes are stable for model and *WN* with respect to embedding dimension and delay. Near the time of the last solar minimum (~2008), the irregularity index exhibits a peak similar to the peaks observed before 1915. This might signal a regime change back from R2 to R1 and the onset of a significant decrease of solar activity.

The book describes the concepts of chaos and order in the "hard" and "soft" sciences.

The authors propose new approach to self-organization of complex distributed systems in logistics. That approach is based on combination of multi-agent paradigm with constraint satisfaction techniques. The proposed solution expresses major features of Swarm Intelligence approach and replaces traditional stochastic adaptation of the swarm of the autonomous agents by constraint-driven adaptation.

The monograph is devoted to the consideration of complex systems from the position of the end the 21st century. The considerable breakthrough in the understanding of complex systems is comprehensively analyzed. Such a breakthrough is connected with the use of the newest methods of nonlinear dynamics, of organization of the modern computational experiments. The book is meant for specialists in different fields of natural sciences and the humanities as well as for all readers who are interested in the recent advancements in science.

This paper revisits the rhetorics of system and irony in Fichte and Friedrich Schlegel in order to theorize the utopic operation and standpoint that, I argue, they share. Both system and irony transport the speculative speaker to the impossible zero-point preceding and suspending the construction of any binary terms or the world itself – a non-place (of the in-itself) that cannot be inscribed into the world’s regime of comprehensibility and possibility. It is because the philosopher and the ironist articulate their speech immanently from this standpoint, that system and irony are positioned as incomprehensible to those framed rhetorically as incapable of occupying it (the dogmatist or the commonsensical public). This standpoint is philosophically important, I maintain, because it allows to think the way the (comprehensibility of the) world is constructed without being bound to the necessity of this construction or having to absolutize, dogmatically, the way things are or can be.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.